1. Field of the Invention
This invention relates to optimal control of a system such as an active noise or vibration control system such as for a helicopter.
2. Background
An active control system consists of a number of sensors which measure the ambient variables of interest (e.g. sound or vibration), a number of actuators capable of generating an effect on these variables (e.g. by producing sound or vibration), and a computer which processes the information received from the sensors and sends commands to the actuators so as to reduce the amplitude of the sensor signals. The control algorithm is the scheme by which the decisions are made as to what commands to the actuators are appropriate.
For tonal control problems, the computation can be performed at an update rate lower than the sensor sampling rate as described in copending application entitled “Computationally Efficient Means for Active Control of Tonal Sound or Vibration”. This approach involves demodulating the sensor signals so that the desired information is near DC (zero frequency), performing the control computation, and remodulating the control commands to obtain the desired output to the actuators. The control computations are therefore performed on the sine and cosine components at the frequency of interest for each sensor signal. These can be represented as a complex variable where the real part is equal to the cosine term, and the imaginary part is equal to the sine term.
The number of sensors is given by ns and the number of actuators is na. The complex harmonic estimator variables that are calculated from the measurements of noise or vibration level can be assembled into a vector of length ns denoted zk at each sample time k. The control commands generated by the control algorithm can likewise be assembled into a vector of length na denoted uk. The commands sent to the actuators are generated by multiplying the real and imaginary parts of this vector by the cosine and sine of the desired frequency.
In the narrow bandwidth required for control about each tone, the transfer function between actuators and sensors is roughly constant, and thus, the system can be modeled as a single quasi-steady complex transfer function matrix, denoted T. This matrix of dimension ns by na describes the relationship between a change in control command and the resulting change in the harmonic estimate of the sensor measurements, that is, Δzk=T Δuk. For notational simplicity, define yk=Δzk, and vk=Δuk. The complex values of the elements of T are determined by the physical characteristics of the system (including actuator dynamics, the structure and/or acoustic cavity, and anti-aliasing and reconstruction filters) so that Tij is the response at the reference frequency of sensor i due to a unit command at the reference frequency on actuator j. Many algorithms may be used for making control decisions based on this model.
The control law is derived to minimize a quadratic performance index
 J=zTWzz+uTWuu+vTWδuv
where Wz, Wu and Wδu are diagonal weighting matrices on the sensor, control inputs, and rate of change of control inputs respectively. A larger control weighting on an actuator will result in a control solution with smaller amplitude for that actuator.
Solving for the control which minimizes J yields:uk+1=uk−Yk(Wuuk+TkTWzzk)whereYk=(TkTWzTk+Wu+Wδu)−1
The matrix Y determines the rate of convergence of different directions in the control space, but does not affect the steady state solution. In the following equation, the step size multiplier β<1 provides control over the convergence rate of the algorithm. A value of approximately β=0.1 may be used, for example.uk+1=uk−βYk(Wuuk+TkHWzzk)
The performance of this control algorithm is strongly dependent on the accuracy of the estimate of the T matrix. When the values of the T matrix used in the controller do not accurately reflect the properties of the controlled system, controller performance can be greatly degraded, to the point in some cases of instability. An initial estimate for T can be obtained prior to starting the controller by applying commands to each actuator and looking at the response on each sensor. However, in many applications, the T matrix changes during operation. For example, in a helicopter, as the rotor rpm varies, the frequency of interest changes, and therefore the T matrix changes. For the gear-mesh frequencies, variations of 1 or 2% in the disturbance frequency can result in shifts through several structural or acoustic modes, yielding drastic phase and magnitude changes in the T matrix, and instability with any fixed-gain controller (i.e. if Tk+1=Tk for all k). Other sources of variation in T include fuel burn-off, passenger movement, altitude and temperature induced changes in the speed of sound, etc.
There are several possible methods for performing on-line identification of the T matrix, including Kalman filtering, an LMS approach, and normalized LMS. For an estimated T matrix, Te, an error vector can be formed asE=y−TevThe estimated T matrix is updated according toTek+1=Tek+EKT
The different estimation schemes differ in how the gain matrix K is selected. The Kalman filter gain K is based on the covariance of the error between T and the estimate Te, given by the matrix P whereM=Pk+QK=Mv/(R+vTMv)Pk+1=M−KvTMand the matrix Q is a diagonal matrix with the same dimension as the number of actuators, and typically with all diagonal elements equal. The scalar R can be set equal to one with no loss in generality provided that the matrices Q and R are constant in time. The normalized LMS approach is very similar, with the gain matrix K given byK=Qv/(1+vTQv)
The algorithm can be used with the Kalman filter approach, or using the normalized LMS approach which is computationally simpler and may provide similar or better performance. The current invention is described in terms of this equation, however, the specific form of the adaptation algorithm is not crucial to the invention.
Any of these adaptation schemes will obtain excellent estimation of the T matrix when there is little noise in the measurements. As noise levels increase, however, there are difficulties as the filter can not distinguish between the effects of noise and the effects of actual changes in T. As a result, the adaptation parameters will tend to drift. Decreasing adaptation gain will decrease the drift but not prevent it, and will degrade the adaptation performance.
The algorithm as described above is self-adaptive in the sense that the plant (i.e., system) model used in the control update calculation is actively updated during closed-loop operation based on changes in the sensor signals resulting from the application of the changes in the actuator commands determined by the control update calculation. However, during steady-state conditions, when changes in the control commands are responding only to “noise” in the estimate of the disturbance being canceled, there is a loss of identifiability of the plant. This loss of identifiability is a result of coupling the control update calculation and the adaptation update calculation together. The adaptation process is estimating the system by observing how changes in actuator commands cause changes in the measured system response; however, due to the control process, the change in control Δu is not independent from the change in measurement Δz. This coupling results in an instability or drift observed in steady state that can be severe if signal-to-noise is poor.
In addition to causing drift behavior, as described above, noise also degrades the quality of the adaptation estimate through two related mechanisms. First, because the signal to noise ratio is reduced the adaptation estimate will include a random noise component. The second effect is more subtle. The adaptation is intended to estimate the transfer function from actuators to sensors at the disturbance frequency. It is reasonable to assume that the physical transfer function does not vary substantially over the bandwidth of the control. Control transients are close to the disturbance frequency, and therefore the transfer functions due to these input signals are close to the desired transfer function. However, the adaptation will also respond to noise on yk and vk that is at higher frequencies (in the demodulated system, i.e. further away from the desired frequency). Changes in the physical system at these frequencies are unknown, however, the most significant change is due to the harmonic estimation filter on yk, the effect of which is known. The estimation of the harmonic components of the sensor signals at the desired frequency requires low-pass filtering to avoid aliasing in downsampling the estimates. While there is no reduction in the amplitude of the sensor information yk in the immediate vicinity of the desired frequency, yk is attenuated at frequencies further form the desired frequency, while vk is not. The algorithm described above will therefore under-predict the desired transfer function because it will average over frequency, and include part of the reduction in amplitude of the harmonic estimator filter. There will be no bias in phase since the noise for positive frequencies should be comparable to the noise for negative frequencies, and the phase effects will therefore cancel.